Let $ψ$ be a conformal map of the unit disk $\mathbb{D}$ onto an unbounded domain and, for $α>0$, let ${F_α}=\left\{ {z \in \mathbb{D}:\left| {ψ\left( z \right)} \right| = α} \right\}$. If ${H^p}\left( \mathbb{D} \right)$ denotes the classical Hardy space and $d_\mathbb{D} {\left( {0,{F_α}} \right)}$ denotes the hyperbolic distance between $0$ and $F_α$ in $\mathbb{D}$, we prove that $ψ$ belongs to ${H^p}\left( \mathbb{D} \right)$ if and only if
\[\int_0^{ + \infty } {α^{p - 1}{e^{ - {d_{\mathbb{D}}}\left( {0,{F_α}} \right)}}dα} < + \infty .\] This result answers a question posed by P. Poggi-Corradini.
Let $ψ$ be a conformal map on $\mathbb{D}$ with $ ψ\left( 0 \right)=0$ and let ${F_α}=\left\{ {z \in \mathbb{D}:\left| {ψ\left( z \right)} \right| = α} \right\}$ for $α>0$. Denote by ${H^p}\left( \mathbb{D} \right)$ the classical Hardy space with exponent $p>0$ and by ${\tt h}\left( ψ\right)$ the Hardy number of $ψ$. Consider the limits $$ L:= \lim_{α\to+\infty}\left( \log ω_{\mathbb D}(0,F_α)^{-1}/\log α\right), \,\, μ:= \lim_{α\to+\infty}\left( d_{\mathbb D}(0,F_α)/\logα\right),$$ where $ω_\mathbb{D}\left( {0,{F_α}} \right)$ denotes the harmonic measure at $0$ of $F_α$ and $d_\mathbb{D} {\le…
We give criteria for the membership of Hankel operators on the Hardy space on the disc in the Dixmier class, and establish estimates for their Dixmier trace. In contrast to the situation in the Bergman space setting, it turns out that there exist Dixmier-class Hankel operators which are not measurable (i.e. their Dixmier trace depends on the choice of the underlying Banach limit), as well as Dixmier-class Hankel operators which do not belong to the $(1,\infty)$ Schatten-Lorentz ideal. A related question concerning logarithmic interpolation of Besov spaces is also discussed.
This paper continues the program initiated in the works by the authors [60], [61] and [62] and by the authors with Li [51] and [52] to establish higher order Poincaré-Sobolev, Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on real hyperbolic spaces using the method of Helgason-Fourier analysis on the hyperbolic spaces. The aim of this paper is to establish such inequalities on the Siegel domains and complex hyperbolic spaces. Firstly, we prove a factorization theorem for the operators on the complex hyperbolic space which is closely related to Geller' operator, as well as the CR inva…
The main purpose of this paper is to discuss Hardy type spaces, Bloch type spaces and the composition operators of complex-valued harmonic functions. We first establish a sharp estimate of the Lipschitz continuity of complex-valued harmonic functions in Bloch type spaces with respect to the pseudo-hyperbolic metric, which gives an answer to an open problem. Then some classes of composition operators on Bloch and Hardy type spaces will be investigated. The obtained results improve and extend some corresponding known results.
Assume that $p\in(1,\infty]$ and $u=P_{h}[ϕ]$, where $ϕ\in L^{p}(\mathbb{S}^{n-1},\mathbb{R}^{n})$. Then for any $x\in \mathbb{B}^{n}$, we obtain the sharp inequalities $$ |u(x)|\leq \frac{\mathbf{C}_{q}^{\frac{1}{q}}(x)}{(1-|x|^2)^{\frac{n-1 }{p}}} \|ϕ\|_{L^{p}} \quad\text{and}\quad |u(x)|\leq \frac{\mathbf{C}_{q}^{\frac{1}{q}} }{(1-|x|^2)^{\frac{n-1 }{p}}} \|ϕ\|_{L^{p} } $$ for some function $\mathbf{C}_{q}(x)$ and constant $\mathbf{C}_{q}$ in terms of Gauss hypergeometric and Gamma functions, where $q$ is the conjugate of $p$. This result generalize and extend some known result from harmoni…
We study a Toeplitz type operator $Q_μ$ between the holomorphic Hardy spaces $H^p$ and $H^q$ of the unit ball. Here the generating symbol $μ$ is assumed to a positive Borel measure. This kind of operator is related to many classical mappings acting on Hardy spaces, such as composition operators, the Volterra type integration operators and Carleson embeddings. We completely characterize the boundedness and compactness of $Q_μ:H^p\to H^q$ for the full range $1<p,q<\infty$; and also describe the membership in the Schatten classes of $H^2$. In the last section of the paper, we demonstrate the usef…
The boundedness and compactness of weighted composition operators on the Hardy space ${\mathcal H}^2$ of the unit disc is analysed. Particular reference is made to the case when the self-map of the disc is an inner function. Schatten-class membership is also considered; as a result, stronger forms of the two main results of a recent paper of Gunatillake are derived. Finally, weighted composition operators on weighted Bergman spaces $\mathcal{A}^2 α(\mathbb{D})$ are considered, and the results of Harper and Smith, linking their properties to those of Carleson embeddings, are extended to this si…
Let ${H^p}\left( \mathbb{D} \right)$ be the Hardy space of all holomorphic functions on the unit disk $\mathbb{D}$ with exponent $p>0$. If $D\ne \mathbb{C}$ is a simply connected domain and $f$ is the Riemann mapping from $\mathbb{D}$ onto $D$, then the Hardy number of $D$, introduced by Hansen, is the supremum of all $p$ for which $f \in {H^p}\left( \mathbb{D} \right)$. Comb domains are a well-studied class of simply connected domains that, in general, have the form of the entire plane minus an infinite number of vertical rays. In this paper we study the Hardy number of a class of comb domain…
We study composition operators on the Fock spaces $\mathcal{F}^2_α(\mathbb{C}^n)$, problems considered include the essential norm, normality, spectra, cyclicity and membership in the Schatten classes. We give perfect answers for these basic properties, which present lots of different characterizations with the composition operators on the Hardy space or the weighted Bergman spaces.
In this paper, we investigate the properties of hyperbolic harmonic mappings in the unit ball $\mathbb{B}^{n}$ in $\IR^n$ $(n\geq 2)$. Firstly, we establish necessary and sufficient conditions for a hyperbolic harmonic mapping to be in the Bloch space $\mathcal{B}(\mathbb{B}^{n})$ and the generalized Bloch space $\mathcal{L}_{\infty,ω}\mathcal{B}_{α,\mathrm{a}}^{0}(\mathbb{B}^{n})$, respectively. Secondly, we discuss the relationship between the integral means of hyperbolic harmonic mappings and that of their gradients. The obtained results are the generalizations of Hardy and Littlewood's rel…
We develop basic properties of solutions to the Dirac-Hodge and Laplace equations in upper half space endowed with the hyperbolic metric. Solutions to the Dirac-Hodge equation are called hypermonogenic functions while solutions to this version of Laplace's equation are called hyperbolic harmonic functions. We introduce a Borel-Pompeiu formula and a Green's formula for hyperbolic harmonic functions. Using a Cauchy Integral formula we are able to introduce Hardy spaces of solutions to the Dirac-Hodge equation. We also provide new arguments describing the conformal covariance of hypermonogenic fu…
We define Hardy spaces $H^p$, $0<p<\infty$, for quasiconformal mappings on the Korányi unit ball $B$ in the first Heisenberg group $\mathbb{H}^1$. Our definition is stated in terms of the Heisenberg polar coordinates introduced by Korányi and Reimann, and Balogh and Tyson. First, we prove the existence of $p_0(K)>0$ such that every $K$-quasiconformal map $f:B \to f(B) \subset \mathbb{H}^1$ belongs to $H^p$ for all $0<p<p_0(K)$. Second, we give two equivalent conditions for the $H^p$ membership of a quasiconformal map $f$, one in terms of the radial limits of $f$, and one using a nontangential …
We characterize the spectra of composition operators on the Hardy space $H^2(B_N)$, when the symbols are elliptic or hyperbolic linear fractional self-maps of $B_N$. Therefore, combining with the result obtained by Bayart \cite{B10}, the spectra of all linear fractional composition operators on $H^2(B_N)$ are completely determined.
In the slice Hardy space over the unit ball of quaternions, we introduce the slice hyperbolic backward shift operator $\mathcal S_a$ with the decomposition process $$f=e_a\langle f, e_a\rangle+B_{a}*\mathcal S_a f,$$ where $e_a$ denotes the slice normalized Szegö kernel and $ B_a $ the slice Blaschke factor. Iterating the above decomposition process, a corresponding maximal selection principle gives rise to the slice adaptive Fourier decomposition. This leads to a adaptive slice Takenaka-Malmquist orthonormal system.
Suppose $α>-1$ and $1\leq p \leq \infty$. Let $f=P_α[F]$ be an $α$-harmonic mapping on $\mathbb{D}$ with the boundary $F$ being absolute continuous and $\dot{F}\in L^p(0,2π)$, where $\dot{F}(e^{iθ}):=\frac{dF(e^{iθ})}{dθ}$. In this paper, we investigate the membership of $f_z$ and $f_{\overline{z}}$ in the space $\mathcal{H}_{\mathcal{G}}^{p}(\mathbb{D})$, the generalized Hardy space. We prove, if $α>0$, then both $f_z$ and $f_{\overline{z}}$ are in $\mathcal{H}_{\mathcal{G}}^{p}(\mathbb{D})$. If $α<0$, then $f_z$ and $f_{\overline{z}}\in \mathcal{H}_{\mathcal{G}}^{p}(\mathbb{D})$ if and only …
The Invariant Subspace Problem ("ISP") for Hilbert space operators is known to be equivalent to a question that, on its surface, seems surprisingly concrete: For composition operators induced on the Hardy space H^2 by hyperbolic automorphisms of the unit disc, is every nontrivial minimal invariant subspace one dimensional (i.e., spanned by an eigenvector)? In the hope of reviving interest in the contribution this remarkable result might offer to the studies of both composition operators and the ISP, I revisit some known results, weaken their hypotheses and simplify their proofs. Sample results…
The Patterson-Sullivan construction is proved almost surely to recover a Bergman function from its values on a random discrete subset sampled with the determinantal point process induced by the Bergman kernel on the unit ball $\mathbb{D}_d$ in $\mathbb{C}^d$. For super-critical weighted Bergman spaces, the interpolation is uniform when the functions range over the unit ball of the weighted Bergman space. As main results, we obtain a necessary and sufficient condition for interpolation of a fixed pluriharmonic function in the complex hyperbolic space of arbitrary dimension (cf. Theorem 1.4 and …
Let $B_n$ denote the unit ball of $\mathbb{C}^n$, $n\ge 1$, and let $\mathcal{D}$ denote a finite product of $B_{n_j}$, $j\ge 1$. Given a non-constant holomorphic function $b: \mathcal{D} \to B_1$, we study the corresponding family $σ_α[b]$, $α\in\partial B_1$, of Clark measures on the distinguished boundary $\partial\mathcal{D}$. We construct a natural unitary operator from the de Branges-Rovnyak space $\mathcal{H}(b)$ onto the Hardy space $H^2(σ_α)$. As an application, for $\mathcal{D}= B_n$ and an inner function $I: B_n \to B_1$, we show that the property $σ_1[I]\llσ_1[b]$ is directly relat…
